Information about Afganistan airspace is published online at www.afgais.com.
Last NOTAM published on OAKX (AFGANISTAN FIR) page reads as follows:
A) OAKX B) 2108160445 C) 2108180445 EST
E) KABUL AIRSPACE HAS BEEN RELEASED TO THE MILITARY. ADVISE TRANSIT AIRCRAFT TO REROUTE. ANY TRANSIT THROUGH KABUL AIRSPACE WILL BE UNCONTROLLED. SURROUNDING FIRS HAVE BEEN ADVISED.)
So, as of 2021-08-16 04:45 UTC [B)2108160445], the civilian air traffic services have not been available for the entire Kabul airspace. The NOTAM does give an estimate of this situation to end at 2021-08-18 04:45 UTC [C) 2108180445 EST], but since it was an estimate, and no new NOTAMs were released since, it must be assumed the situation has persisted.
There most certainly was some sort of ATC for Hamid Karzai International, but most likely this was provided by military personnel, and the last flight quite obviously took these controllers with them.
As the Taliban took over the control of the capital, effectively any judicial and executive entities related to former governing structures ceased to exist. Air traffic services were delivered under proxy from the former government, and thus could not be lawfully provided anymore (Chicago convention and international law may have something to say about this, but it's well outside my expertise). And, I bet the civilian controllers had other things to worry about aside the legal considerations of ATC services. Hence the surrendering the airspace to military. Interesting side question is, to which military...
Now, as for operating the last flight, it simply departed as any flight from an uncontrolled airport/airfield. These are not uncommon operations for military aviators, especially those at airlift duties. Flight coordination was most likely provided by an airborne control center. Overall it is imperative that any military operation is as redundant as possible, not requiring any kind of fixed or permanent units to maintain them.
Addendum: Many thanks for @uhoh for referring the following link to a news article describing the last phases of airlift operation from Kabul. It is for sure a condensed and somewhat dramatized version of how thing actually went, but essentially the last flights departed in a remotely coordinated manner with preplanned sequencing. The remote coordination was, as I previously mentioned, not necessary in any sense, so should there have been any lapse in communications, the departure would have proceeded as it did regardless of any thumb-upping or lack thereof.
CNN: Air Force commander details final hours before last US planes left Afghanistan
It must have been quite a sight having five C-17's execute a tactical takeoff within only three minutes.